A Fast Algorithm for Quadrature by Expansion in Three Dimensions

TL;DR

This paper introduces a fast, scalable algorithm combining Quadrature by Expansion with a modified Fast Multipole Method for efficient evaluation of layer potentials in three dimensions, supported by numerical experiments.

Contribution

It extends a 2D QBX-FMM formulation to 3D, providing a rigorous, high-order, accelerated quadrature scheme for layer potential evaluation.

Findings

Achieves high accuracy in 3D layer potential evaluation

Demonstrates improved performance and scalability

Validates effectiveness through numerical experiments on Laplace and Helmholtz equations

Abstract

This paper presents an accelerated quadrature scheme for the evaluation of layer potentials in three dimensions. Our scheme combines a generic, high order quadrature method for singular kernels called Quadrature by Expansion (QBX) with a modified version of the Fast Multipole Method (FMM). Our scheme extends a recently developed formulation of the FMM for QBX in two dimensions, which, in that setting, achieves mathematically rigorous error and running time bounds. In addition to generalization to three dimensions, we highlight some algorithmic and mathematical opportunities for improved performance and stability. Lastly, we give numerical evidence supporting the accuracy, performance, and scalability of the algorithm through a series of experiments involving the Laplace and Helmholtz equations.